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We obtain new inequalities for the Fourier transform, both on Euclidean space, and on non-compact, rank one symmetric spaces. In both cases these are expressed as a gauge on the size of the transform in terms of a suitable integral modulus of continuity of the function. In all settings, the results present a natural corollary: a quantitative form of the Riemann-Lebesgue lemma. A prototype is given in one-dimensional Fourier analysis.

Contents 1. A general criterion for pointwise Fourier inversion 2. Pointwise Fourier inversion on R n (n = 3) 3. Fourier inversion on R 2 4. Fourier inversion on R n (general n) 5. Fourier inversion on spheres 6. Fourier inversion on complex projective space, and variants 7. Fourier inversion on hyperbolic space, and variants 8. Fourier inversion on strongly scattering manifolds 9. Hermite expansions and the Schr odinger equation 10. Nonspherical Fourier inversion on R n 11. Gibbs phenomena on manifolds A. The Dirichlet kernel and the wave equation B. The heat kernel and the wave k ernel C. Distributions oscillatory at the origin

Estimation of solution norms and stability for time-dependent nonlinear systems is ubiquitous in numerous engineering, natural science and control problems. Yet, practically valuable results are rare in this area. This paper develops a novel approach, which bounds the solution norms, derives the corresponding stability criteria, and estimates the trapping/stability regions for some nonautonomous and nonlinear systems, which arise in various application domains. Our inferences rest on deriving a scalar differential inequality for the norms of solutions to the initial systems. Utility of the Lipschitz inequality linearizes the associated auxiliary differential equation and yields both the upper bounds for the norms of solutions and the relevant stability criteria. To refine these inferences, we introduce a nonlinear extension of the Lipschitz inequality, which improves the developed bounds and allows estimation of the stability basins and trapping regions for the corresponding systems. Finally, we confirm the theoretical results in representative simulations.

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